Benchmark solution
=====================


Calculation method used for the reference solution
--------------------------------------------------------

Analytical solution.

Temperature varying linearly according to :math:`\mathrm{CD}`

Isotherms parallel to faces :math:`\mathrm{CHKF}` and :math:`\mathrm{DIJE}`.

In the frame: :math:`(\frac{\mathrm{CD}}{\parallel \mathrm{CD}\parallel },\frac{\mathrm{CH}}{\parallel \mathrm{CH}\parallel },\frac{\mathrm{CF}}{\parallel \mathrm{CF}\parallel },)`, we have:

:math:`\left[\begin{array}{}{\varphi }_{x}\\ {\varphi }_{y}\\ {\varphi }_{z}\end{array}\right]=\left[\begin{array}{}-({\lambda }_{X}{\mathrm{cos}}^{2}\alpha +{\lambda }_{Y}{\mathrm{sin}}^{2}\alpha )\frac{\partial T}{\partial x}\\ -({\lambda }_{X}-{\lambda }_{Y})\mathrm{cos}\alpha \mathrm{sin}\alpha \frac{\partial T}{\partial x}\\ 0\end{array}\right]`

with:

:math:`{\varphi }_{x}=1200{\varphi }_{y}=400\alpha =({X}_{\mathrm{1,}}\mathrm{CD})` :math:`T(x)=\frac{-{\varphi }_{x}}{{\lambda }_{{X}_{1}}{\mathrm{cos}}^{2}\alpha +{\lambda }_{Y}{\mathrm{sin}}^{2}\alpha }x+T(A)`

That is: :math:`T(x)=-\mathrm{1600.x}+20.`

If :math:`\beta =(\mathrm{CD},{X}_{0})`: :math:`\begin{array}{cc}\varphi \mathrm{.}{X}_{0}=\mathrm{cos}\beta \mathrm{.}{\varphi }_{X}-\mathrm{sin}\beta {\varphi }_{y}& \text{soit}720\\ \varphi \mathrm{.}{Y}_{0}=\mathrm{sin}\beta \mathrm{.}{\varphi }_{X}+\mathrm{cos}\beta {\varphi }_{L}& \text{soit}720\end{array}`


Benchmark results
----------------------

Temperature at points :math:`A,B,G`.

Flow following directions :math:`{X}_{0}` and :math:`{Y}_{0}`.

:math:`T(A)=100T(B)=20T(G)=60\Phi \mathrm{.}{X}_{0}=720\Phi \mathrm{.}{Y}_{0}=1040`


Bibliographical references
---------------------------


1. N. RICHARD: Technical note HM-18/94/0011, "Development of thermal anisotropy in *Aster* software".